338 DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 
The equation between the amplitudes and 
tan('4'—pf')=J tan/u^, gives 
(1 +j)sinju, cos|x 
tan^/: 
cos^ —j 
(91.) 
Eliminating %}/ by the help of this equation, from the value of tanr given in the pre- 
ceding group, 
tanr: 
(1 — y) sinju. cos|x ^ ^ cos^/u, sin^/x 
■ X 
cos^f4— y sin^fi 
Using this transformation and reducing, 
tan(r-l-r') = tan/!>t->/l — sin^/U;, (92.) 
a simple expression for the length of the tangent arc to the spherical parabola be- 
tween the perpendicular arcs let fall from the centre and focus upon it. 
From the last equation we may derive 
^ sin^fx. 
cos^jx — 7“^ sinV 
Using the preceding transformations, we may show that 
(93.) 
Hence 
Therefore (86.) becomes 
1 
^^^0 2 sinju. cosjx v'l — sin^^x 
cos‘*/x sin‘‘|x 
©=2(r-|-r') (94.) 
,0 
dvj/ 
'• d\I/ 
'l+ 
sin^vl/ 
\/i- 
sin^\|/ 
1 
1 
> 
u+y) 
' sin^4' 
('+i)2=(i+y)(’-+^)- (M. 
We have thus shown that in the particular case of the general formula for com- 
paring elliptic functions of the third order with reciprocal parameters, when the 
parameter is positive and equal to the modulus, the circular arc in the formula of com- 
parison (87.) is equal to twice the arc of the great circle touching the curve and inter- 
cepted between the perpendicular arcs let fall from the centre and focus upon it. 
If we take the parameter with a negative sign, the circular arc r in (62.) will re- 
present the tangent arc between the point of contact and the foot of the focal perpen- 
dicular. 
The spherical parabola, like any other spherical ellipse, may be considered as the 
intersection of an elliptic cylinder with a sphere whose centre is on the axis of the 
cylinder. 
Let a and h be the semiaxes of the base of the cylinder, and k the radius of the 
sphere, a and (3 being the principal semiarcs of the spherical parabola, 
tan^a: 
*2 
tan^(3=T^. 
but in (59*.) we found tan^a— tan^j3= 1 ; hence substituting, 
k‘^=a^{\-\-i) 
( 96 .) 
